Imagine that you want to compare the standard deviations of two sample
to determine if they differ in any significant way, in this situation you
use the F distribution and perform an F-test. This situation commonly occurs
when conducting a process change comparison: "is a new process more
consistent that the old one?".

In this example we'll be using the data for ceramic strength from http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm.
The data for this case study were collected by Said Jahanmir of the NIST
Ceramics Division in 1996 in connection with a NIST/industry ceramics consortium
for strength optimization of ceramic strength.

The alternative hypothesis: there is a difference in means (two
sided test)

Reject if F(1-alpha/2; N1-1, N2-1) <= F <= F(alpha/2; N1-1,
N2-1)

The alternative hypothesis: Standard deviation of sample 1 is
greater than that of sample 2

Reject if F < F(alpha; N1-1, N2-1)

The alternative hypothesis: Standard deviation of sample 1 is
less than that of sample 2

Reject if F > F(1-alpha; N1-1, N2-1)

Where F(1-alpha; N1-1, N2-1) is the lower critical value of the F distribution
with degrees of freedom N1-1 and N2-1, and F(alpha; N1-1, N2-1) is the upper
critical value of the F distribution with degrees of freedom N1-1 and N2-1.

The upper and lower critical values can be computed using the quantile
function:

In this case we are unable to reject the null-hypothesis, and must instead
reject the alternative hypothesis.

By contrast let's see what happens when we use some different sample
data:, once again from the NIST Engineering Statistics Handbook:
A new procedure to assemble a device is introduced and tested for possible
improvement in time of assembly. The question being addressed is whether
the standard deviation of the new assembly process (sample 2) is better
(i.e., smaller) than the standard deviation for the old assembly process
(sample 1).

In this case we take our null hypothesis as "standard deviation 1
is less than or equal to standard deviation 2", since this represents
the "no change" situation. So we want to compare the upper critical
value at alpha (a one sided test) with the test statistic,
and since 3.35 < 3.6 this hypothesis must be rejected. We therefore
conclude that there is a change for the better in our standard deviation.